To simplify the expression [tex]\(\frac{y}{\sqrt{8} - \sqrt{18} + \sqrt{12} + \sqrt{50} - \sqrt{106}}\)[/tex], let's go through the steps to reduce it into a simpler form.
1. Identify Terms:
The terms in the denominator are:
- [tex]\(\sqrt{8}\)[/tex]
- [tex]\(-\sqrt{18}\)[/tex]
- [tex]\(\sqrt{12}\)[/tex]
- [tex]\(\sqrt{50}\)[/tex]
- [tex]\(-\sqrt{106}\)[/tex]
2. Attempt to Simplify Each Square Root Term:
- [tex]\(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}\)[/tex]
- [tex]\(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\)[/tex]
- [tex]\(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\)[/tex]
- [tex]\(\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}\)[/tex]
- [tex]\(\sqrt{106}\)[/tex] remains as it is since it's already in simplest form.
3. Substitute Simplified Terms Back Into the Expression:
Substituting the simplified terms, the expression becomes:
[tex]\[
\frac{y}{2\sqrt{2} - 3\sqrt{2} + 2\sqrt{3} + 5\sqrt{2} - \sqrt{106}}
\][/tex]
4. Combine Like Terms:
Combine the terms involving [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
2\sqrt{2} - 3\sqrt{2} + 5\sqrt{2} = (2 - 3 + 5)\sqrt{2} = 4\sqrt{2}
\][/tex]
Now, the expression in the denominator is:
[tex]\[
4\sqrt{2} + 2\sqrt{3} - \sqrt{106}
\][/tex]
5. Rewrite the Expression as One Term:
Hence, the simplified form of the original expression is:
[tex]\[
\frac{y}{4\sqrt{2} + 2\sqrt{3} - \sqrt{106}}
\][/tex]
So, the simplified form of [tex]\(\frac{y}{\sqrt{8} - \sqrt{18} + \sqrt{12} + \sqrt{50} - \sqrt{106}}\)[/tex] is:
[tex]\[
\boxed{\frac{y}{- \sqrt{106} + 2\sqrt{3} + 4\sqrt{2}}}
\][/tex]
